I know there must be something unmathematical in the following but I don't know where it is: −1−−−√1−1−−−√1√−1−−−√1−1−−−√−11−−−√−1−−−√ii2−1=i=1i=1i=1i=1i=1i=1i=1=1!!!

Between your third and fourth lines, you use a√b√=ab−−√. This is only (guaranteed to be) true when a≥0 and b>0.
edit: As pointed out in the comments, what I meant was that the identity a√b√=ab−−√ has domain a≥0 and b>0. Outside that domain, applying the identity is inappropriate, whether or not it "works."
In general (and this is the crux of most "fake" proofs involving square roots of negative numbers), x√ where x is a negative real number (x<0) must first be rewritten as i|x|−−√ before any other algebraic manipulations can be applied (because the identities relating to manipulation of square roots [perhaps exponentiation with non-integer exponents in general] require nonnegative numbers).
This similar question, focused on −1=i2=(−1−−−√)2=−1−−−√−1−−−√=!−1⋅−1−−−−−−√=1√=1, is using the similar identity a√b√=ab−−√, which has domain a≥0 and b≥0, so applying it when a=b=−1 is invalid.